Solvable model for pair excitation in trapped Boson gas at zero temperature

TL;DR

This paper develops an exact solvable model for pair excitations in a trapped Bose-Einstein condensate at zero temperature, analyzing the dynamics of particle pairs transitioning from the condensate state.

Contribution

It introduces a nonlocal equation for pair excitation functions and provides an asymptotic solution using special functions under a slowly varying potential.

Findings

Derived an explicit solution for pair excitation function K0.

Analyzed asymptotic behavior for large times and distances.

Connected pair excitations to the nonlinear Schrödinger equation.

Abstract

In Bose-Einstein condensation, a macroscopically large number of particles occupy the same single-particle quantum state. Our goal is to study time-dependent aspects of particle excitations to states other than the single-particle macroscopic state in trapped dilute atomic gases. We adopt the view that atoms are excited in pairs so that their scattering from the single-particle state to vector positions x and y at time t is described by the pair-excitation function, K0(x,y,t). We solve a nonlocal equation for K0 under a slowly varying external potential by assuming that the wave function of the macroscopic state satisfies a time-independent nonlinear Schroedinger equation. For zero initial excitation (K0=0 at t=0) and sufficiently large t, we evaluate asymptotically K0 in terms of the one-variable Lommel function for any distance |x-y|.